Extremal results for graphs of bounded metric dimension

Geneson, Jesse; Kaustav, Suchir; Labelle, Antoine

doi:10.1016/j.dam.2021.11.015

Cited by 7 publications

(3 citation statements)

References 16 publications

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“…A new parameter related to metric dimension introduced in [ 17 ], few generalized results of metric dimension in terms of diameter regarding the bipartite type of graphs found in [ 18 ]. Extremal results in terms of bounded metric dimension are discussed in [ 19 ]. A weighted metric dimension is introduced in [ 20 ], they discussed a few hard and easy cases to tackle and also linked this to the basic metric dimension.…”

Section: Introductionmentioning

confidence: 99%

Edge based metric dimension of various coffee compounds

Ahmad,

Koam,

Azeem

et al. 2024

PLoS ONE

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An important dietary source of physiologically active compounds, coffee also contains phenolic acids, diterpenes, and caffeine. According to a certain study, some coffee secondary metabolites may advantageously modify a number of anti-cancer defense systems. This research looked at a few coffee chemical structures in terms of edge locating numbers or edge metric size to better understand the mechanics of coffee molecules. Additionally, this research includes graph theoretical properties of coffee chemical structures. The chemicals found in coffee, such as caffeine, diterpene or cafestol, kahweol, chlorogenic, caffeic, gallotannins, and ellagitannins, are especially examined in these publications.

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Section: Introductionmentioning

confidence: 99%

Edge based metric dimension of various coffee compounds

Ahmad,

Koam,

Azeem

et al. 2024

PLoS ONE

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show abstract

“…Applications of metric dimensions to various fields, e.g., navigation of robots [7], chemistry [8,9], coin-weighing, and mastermind game [10] have been presented in the literature. Further studies on metric dimension and metric basis were conducted in [11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Metric Dimensions of Bicyclic Graphs

et al. 2023

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The distance d(va,vb) between two vertices of a simple connected graph G is the length of the shortest path between va and vb. Vertices va,vb of G are considered to be resolved by a vertex v if d(va,v)≠d(vb,v). An ordered set W={v1,v2,v3,…,vs}⊆V(G) is said to be a resolving set for G, if for any va,vb∈V(G),∃vi∈W∋d(va,vi)≠d(vb,vi). The representation of vertex v with respect to W is denoted by r(v|W) and is an s-vector(s-tuple) (d(v,v1),d(v,v2),d(v,v3),…,d(v,vs)). Using representation r(v|W), we can say that W is a resolving set if, for any two vertices va,vb∈V(G), we have r(va|W)≠r(vb|W). A minimal resolving set is termed a metric basis for G. The cardinality of the metric basis set is called the metric dimension of G, represented by dim(G). In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension.

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“…Hayat et al [9] determined the exact values of metric dimensions of multi-partite graphs, effectively generalizing the already established results of bipartite graphs. For further studies on metric dimensions, we refer the reader to [10][11][12][13][14][15][16][17][18] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Constant Time Calculation of the Metric Dimension of the Join of Path Graphs

et al. 2023

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The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length of the shortest path from u to v and is always symmetrical. An ordered subset W=w1,w2,w3,⋯,wk of V(G) is a resolving set for G, if for ∀u,v∈V(G), there exists wi∈W ∋ d(u,wi)≠d(v,wi). A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as dim(G). For the graph G1=(V1,E1,) and G2=(V2,E2), their join is denoted by G1+G2. The vertex set of G1+G2 is V1∪V2 and the edge set is E=E1∪E2∪uv,u∈V1,v∈V2. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.

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Extremal results for graphs of bounded metric dimension

Cited by 7 publications

References 16 publications

Edge based metric dimension of various coffee compounds

Edge based metric dimension of various coffee compounds

Metric Dimensions of Bicyclic Graphs

Constant Time Calculation of the Metric Dimension of the Join of Path Graphs

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